Help with commutators

1. The problem statement, all variables and given/known data
(H - hamiltonian, P - momentum, Q- position)
Given two operators Q and Q', I have shown that [H(P, Q), Q] = [H(P, Q), Q']. I was wondering if this meant that I could assume that an energy spectrum found from H(P, Q) could be related to that of H(P, Q'). I am under the impression that the spectrum would be shifted over by a constant (multiplied by identity).

2. Relevant equations
Given above.

3. The attempt at a solution
The above problem comes from my attempt to solve a more complicated problem.